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The Calculation of Chemical Potentials in Natural Waters. Application to Mixed Chloride-Sulfate Solutions G. MICHEL LAFON Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, Md. 21218
A fundamental problem in the study of natural waters is ascertaining the chemical potentials of the dissolved components. The chemical potentials are of particular interest because they provide the most natural expression of equilibrium between the solution and other phases, their gradients are the driving forces of molecular diffusion, and they also characterize the free energy available for non-equilibrium processes such as the precipitation and dissolution of minerals. Because many of the solid phases in contact with natural waters are reactive over relatively short periods of time, departures from equilibrium are often small. For example, the calcite saturation factor in the deep ocean is about 0.6 to 0.7 and decreases below 0.5 only very rarely (1) . The barium concentrations of sediment pore waters from the eastern Pacific Ocean appear to be controlled by equilibrium with barite (2). To study equilibrium and non-equilibrium processes in natural waters, we need to determine the chemical potentials of dissolved components with an accuracy sufficient to detect these small departures from equilibrium (say, 10 per cent or better). The chemical potentials of dissolved salts can be expressed either in terms of the activities of constituent ions, or more directly in terms of the activities of neutral components.These two approaches have recently been discussed by Leyendekkers (3, 4) and Whitfield (5), and Millero (6) has reviewed their application to sea water. Results for metal ions, chloride and bicarbonate are in good agreement, but there are serious discrepancies for sulfate and carbonate. Ionic activities cannot be measured directly and must instead be estimated using some arbitrary nonthermodynamic assumption.The models used to estimate ionic activities in mixed-electrolyte solutions (e.g. ion-pairing or specific interaction models) fail in some cases where there are strong interactions between ions (7). Thus, the choice of neutral components which can be investigated experimentally appears preferable. Many empirical models of the chemical potentials of neutral components in mixed-electrolyte solutions have been proposed, and they are discussed in some detail by Harned and Owen (8), Robinson 97 Church; Marine Chemistry in the Coastal Environment ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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and Stokes (9) and Hamed and Robinson (10) . More recently, Friedman (11) has suggested a generalization of Young's (12) Rule, proposing that the excess free energy of a mixed-electrolyte solution be expressed as the sum of the contributions of single-salt solutions at the same ionic strength and of an expansion i n the products of the ionic strength fractions of each electrolyte. Using Friedman's approach, Reilly and Wood (13) and Scatchard and colleagues (14,15) have obtained equations for the free energy of arbitrary mixed-electrolyte solutions which appear to be superior to previous treatments. The purpose of this paper i s to show that the equations of Reilly, Wood and Robinson (16) accurately predict the values of chemical potentials i n mixed-electrolyte solutions where there are strong ion-ion interactions. The additivity of the interaction terms i n the general equations of (16) i s demonstrated for systems of the type NaCl-MS0t»-H20. Thus, i t i s l i k e l y that these equations predict chemical potentials i n natural waters more accurately than has been possible previously. Chemical Potentials and Interaction Parameters i n mixed-electrolyte solutions. Using Reilly and Wood's (13) expression for the free energy of an arbitrary mixed-electrolyte solution, Reilly, Wood and Robinson (16) have derived expressions for the osmotic coefficient and for the activity coefficients of components i n terms of the experimental properties of single-salt solutions and of interaction parameters. These parameters, denoted here by #(1,J,K) can be obtained from measurements i n electrolyte mixtures with a common ion, IJ-IK. Although the general equations of Reilly,Wood and Robinson (RWR) are cumbersome, their mathematical treatment is simple and easily programmed on a computer. Because relatively few single-salt data and interaction parameters are needed for the computation of a l l the chemical potentials i n any mixed-electrolyte solution, the approach of R e i l l y , Wood and Robinson i s particularly well suited to the study of natural waters, especial l y sea water (17). Scatchard and colleagues (14,15) have presented a very similar treatment, but have chosen to represent the properties of single-salt solutions and the interaction parameters by power series i n the ionic strength. The RWR equations have the advantage that they use experimental quantities directly and that they do not presuppose the functional form of the interaction parameters. Friedman (11) has shown that, for asymmetrically charged mixtures, the interaction parameter approaches In I i n the limit of i n f i n i te dilution, which makes a power function representation highly doubtful at low ionic strength. The RWR expressions for a c t i v i t y coefficients are functions of the interaction parameters and, independently, of their part i a l derivatives with respect to the ionic strength. The
Church; Marine Chemistry in the Coastal Environment ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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expression for the osmotic coefficient depends solely on the quantity 8 ( J ^ ) / 3 J which can be conveniently denoted by the symbol w. To achieve a consistent thermodynamic description of mixtures with a common ion, the values of g obtained from a c t i v i t y c o e f f i cients must be related to those of w obtained from the osmotic coefficient. The relation between g and w i s discussed i n detail elsewhere (18). We simply r e c a l l here that, i f w can be represented empirically by some function / over the ionic strength interval: x i < J = "
4
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9
4
^
The equilibrium constant at 25 C was calculated from the s o l u b i l i ty of gypsum i n water and estimates of the activity and osmotic coefficients of CaSOi* i n dilute solutions. Although no data for CaSOi* are available, Harned and Owen (8) have pointed out that the activity and osmotic coefficients of the divalent metal sulfates depend very l i t t l e on the nature of the metal ion at low ionic strength. Therefore, the properties of ZnSOi* solutions were used as good approximations of those of CaSCH solutions. Note that the equilibrium constant i n equation 15 differs markedly from the limiting s o l u b i l i t y extrapolated to zero ionic strength by Marshall and Slusher (24). Their extrapolation requires that the activity coefficient of CaS0t» l i e above the Debye-HUckel limiting slope at low ionic strength, while we know that the a c t i v i t y coeff i c i e n t of ZnSOj* approaches this slope from below.Moreover, conductance curves for 2:2 electrolytes are also known to approach the limiting law from below (8), a fact consistent with the widespread assumption of ion-pairing i n these solutions. Thus, the limiting s o l u b i l i t i e s extrapolated by Marshall and Slusher (24) are probably not good estimates of the equilibrium constant for gypsum dissolution. The results derived from the s o l u b i l i t y of gypsum are quite similar to those obtained above from that of barite. Therefore, only sample calculations at 3.0 and 6.0 m are reported here. At these ionic strengths, the logarithms of the molal s o l u b i l i t i e s are -1.240 and -1.305 respectively [equation 6 of reference (24)]. The corresponding values of the trace activity coefficient of CaSO^ are 0.0982 and 0.1341, leading to values of [#(Na,Cl,S0. ) + #(Na,Ca,Cl)] of -0.0692 at 3.0 m and -0.0495 at 6.0 m. Next, we obtain #(Na,Ca,Cl) from the thermodynamic properties of mixed NaCl-CaCl solutions i n a manner similar to that used for ) + ^(Na,Ca,Cl)] calculated from the data for common-ion mixtures are -0.0640 at 3.0 m and -0.0495 at 6.0 m.The corresponding values obtained from the s o l u b i l i t y of gypsum are -0.0692 and -0.0570 respectively. Here, again, the agreement between the two methods i s excellent, especially when we consider the experimental uncertainties involved at each step of the calculations. We conclude that the s o l u b i l i t y of gypsum can be predicted accurately by equation 3 and the interaction parameters given by equations 10 and 17. Conclusions We have shown that the osmotic and a c t i v i t y coefficients i n NaCl-BaCl2-H 0 and NaCl-CaCl -H 0 can be described accurately by the equations of Reilly, Wood and Robinson (16) where the interaction parameters are given by equations 12, 14, 16 and 17. We have further shown that these interaction parameters can be added to the parameter that characterizes NaCl-Na S0i -H 0 to provide accurate estimates of the trace a c t i v i t y coefficients of BaSCKand CaSOif i n sodium chloride solutions. Thus, i t appears that the additive character of the interaction terms i n the general equations of Reilly, Wood and Robinson i s v a l i d even for systems where marked specific interactions between ions are known to exist (in this case, between metal ions and sulfate). This strong ly suggests that the general RWR equations can be used to estimate chemical potentials i n natural waters more accurately than has been possible to date. More s p e c i f i c a l l y , these equations together with appropriate expressions for the interaction parameters are l i k e l y to y i e l d significantly improved estimates of the a c t i v i t y coefficients of sulfate and carbonate components i n sea water and i n continental brines. Finally, the s o l u b i l i t i e s of sparingly soluble sulfate salts i n sodium chloride solutions can be used in conjunction with values of #(Na,Cl,S0O to estimate 2
2
2
2
t
2
Church; Marine Chemistry in the Coastal Environment ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
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the interaction parameter ^(Na,M,Cl) in systems for which no direct data are available. Literature Cited (1) Takahashi T., in "Dissolution of Deep-Sea Carbonates",(1975) Cushman Foundation for Foraminiferal Research, Spec. Publ. 13, p. 11, Washington. (2) Church T. M. and Wohlgemuth K., Earth Planet. Sci. Letters (1972) 15, 35. (3) Leyendekkers J. V., Anal. Chem. (1971) 43, 1835. (4)__________________,Marine Chemistry (1973) 1, 75. (5) Whitfield M. , Marine Chemistry (1973) 1, 251. (6) Millero F. J., in "Annual Review of Earth and Planetary Sciences", vol. 2 (1974), p. 101. (7) Lafon G. M. and Truesdell A. H., ms in preparation. (8) Harned H. S. and Owen B. B., "The Physical Chemistry of Electrolytic Solutions", 3rd Edition, xxxiii + 803 p., Reinhold Book Corp., New York (1958). (9) Robinson R. A. and Stokes R. H.,"Electrolyte Solutions", 2nd Edition(revised), xv + 571 p., Butterworths, London,(1970). (10) Harned H. S. and Robinson R. A., "Multicomponent Electrolyte Solutions", xiii + 110 p., Pergamon Press, Oxford, (1968). (11) Friedman H. L., J. Chem. Phys. (1960) 32, 1351. (12) Young T. F. and Smith M. B., J. Phys. Chem. (1954) 58, 716. (13) Reilly P. J. and Wood R.H.,J. Phys. Chem. (1969) 73, 4292. (14) Scatchard G., J. Am. Chem. Soc. (1961) 83, 2636. (15) Scatchard G., Rush R. M. and Johnson J. S., J. Phys. Chem. (1970) 74, 3786. (16) Reilly P. J., Wood R. H. and Robinson R. A., J. Phys. Chem. (1971) 75, 1305. (17) Robinson R. A. and Wood R. H., J. Solution Chem. (1972) 1,481 (18) Lafon G. M., ms in preparation. (19) Templeton C. C., J. Chem. Eng. Data (1960) 5, 514. (20) Rosseinsky D. R., Trans. Faraday Soc. (1958) 54, 116. (21) Robinson R. A. and Bower V. E., J. Res. Nat. Bur. Standards (1965) 69A, 19. (22) Lanier R. D., J. Phys. Chem. (1965) 69, 3992. (23) Christenson P. G., J. Chem. Eng. Data (1973) 18, 286. (24) Marshall W. L. and Slusher R., J. Phys. Chem. (1966) 70, 4015 (25) Robinson R. A. and Bower V. E., J. Res. Nat. Bur. Standards (1966) 70A, 313.
Church; Marine Chemistry in the Coastal Environment ACS Symposium Series; American Chemical Society: Washington, DC, 1975.
